In the field of theoretical cosmology, researchers have long struggled to devise models for how matter and energy behave as the universe expands.
Brown researchers recently showed that mathematical structures known as cluster algebras can help describe how fluctuations in the early universe relate to one another, a finding that may help in solving the modeling problem. A mathematical technique built on these structures, known as a cluster bootstrap, has the potential to simplify complex cosmological calculations and offer a better understanding of how the universe formed.
According to Shruti Paranjape, the paper’s first author and a postdoctoral research associate in the Department of Physics, this new research grew out of previous work on scattering amplitudes, which physicists calculate to describe what happens when two particles collide. Scatting amplitudes give the probabilities associated with the different possibilities of what might happen when two things scatter.
“All of us in this collaboration, we largely study the mathematical or theoretical properties of scattering amplitudes,” Paranjape said. “Recently, what we noticed was that a lot of these mathematical properties of scattering amplitudes extend to these other observables.”
Observables are measurable quantities in particle physics, such as position, velocity, spin and energy.
In this study, the researchers looked at the statistical relationships between fluctuations in quantum fields, which underlie the particles in the universe and describe their properties across different locations and times. These relationships are described by “cosmological correlators,” Anastasia Volovich, one of the paper’s principal investigators and a professor of physics, wrote in an email to The Herald.
“The basic idea is when you look out at the universe, matter is very evenly distributed, but not completely evenly distributed. There are clumps of galaxies, and it’s a long-standing problem in cosmology to explain exactly why there’s a certain amount of clumpiness,” said Marcus Spradlin, another author of the paper and a professor of physics.
Spradlin offered an example of how to visualize the concept of cosmological correlation: If you spot a galaxy at a specific position in the sky, you can calculate the probability of seeing a second galaxy from any given angle away from the first.
According to Paranjape, theoretical and observational physicists have to work together to develop models by aligning experimental observation with mathematical theory.
Observational cosmologists look for the radiation left over from the Big Bang, which can provide insight into the universe’s formation and its current expansion. Once they extract the data, they compare it against fundamental theory, giving theoretical physicists a basis to work from.
On the theoretical end, Paranjape explained that researchers start by using a “toy model” — “a theoretical setup, which is very easy to study.” Starting with some preliminary theory allows theorists to tweak it as they go, using data from observers. Once the theorists develop a model, observational physicists can use the new methods to collect more data, which they then send back to the theoreticalists to verify their models.
Paranjape emphasized that for this study, researchers had to “brute force” the calculations to find the structure. Now that the researchers have established this bootstrap model for cosmology, future scientists can lean on these new methods to conduct their calculations.
Spradlin explained that this bootstrap method “inputs information about what kind of properties the answer should have to try to find a shortcut to the answer without having to do a laborious calculation.”
Once these calculations are complete and researchers have a sense of what the broad structure might be, they can use the model to make predictions and then assess if those predictions are correct.
“Sometimes you can go one step further, and you can actually prove that the structure will hold,” Paranjape added. “And in our case, that’s what we did.” The researchers showed that cosmological wave functions can be defined by a specific cluster algebra.
“Once you see that the thing or the object you’re studying satisfies some rigid mathematical properties, it helps you carry out calculations that were previously impossible,” Spradlin added.
Alice Xie is a section editor for Science and Research from Los Angeles, California. She studies Applied Mathematics and Biology, and enjoys reading gut wrenching literature in her free time.
